Solve (x-7)^2 = 36: Find X Values

Hey guys! Let's dive into a fun math problem today where we'll be solving a quadratic equation. Specifically, we're going to tackle the equation $(x-7)^2 = 36$. This type of problem is super common in algebra, and mastering it will definitely boost your math skills. So, let's break it down step by step and make sure we understand exactly how to find the values of x.

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's quickly recap what quadratic equations are all about. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (in our case, x) is 2. The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants. However, our equation, $(x-7)^2 = 36$, is presented in a slightly different form, which we'll address shortly.

Why are Quadratic Equations Important?

You might be wondering, "Why should I care about quadratic equations?" Well, they pop up all over the place in real-world applications! From calculating the trajectory of a projectile (like a ball being thrown) to designing structures and modeling various physical phenomena, quadratic equations are essential tools. They help us describe curves and parabolic paths, making them incredibly useful in physics, engineering, and even economics.

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, each with its own strengths. The most common ones include:

1. Factoring: This method involves breaking down the quadratic expression into two binomial factors. It's a quick method when the equation can be factored easily.
2. Completing the Square: This technique involves manipulating the equation to form a perfect square trinomial, which can then be easily solved.
3. Quadratic Formula: This is a universal method that works for any quadratic equation. The formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
4. Taking the Square Root: This method is particularly useful when the equation is in the form $(x - h)^2 = k$, which is exactly the form of our equation today!

Solving (x-7)^2 = 36 by Taking the Square Root

Okay, let's get back to our equation: $(x-7)^2 = 36$. Since it's already in the form of a squared term equal to a constant, the easiest way to solve it is by taking the square root of both sides. This method is straightforward and efficient, minimizing the steps needed to find the solutions.

Step 1: Take the Square Root of Both Sides

The first step is to take the square root of both sides of the equation. Remember, when we take the square root, we need to consider both the positive and negative roots. This is because both the positive and negative values, when squared, will give us the same positive result. So, we have:

$\sqrt{(x-7)^2} = \pm \sqrt{36}$

This simplifies to:

$x - 7 = \pm 6$

Step 2: Solve for x

Now we have two separate equations to solve:

1. $x - 7 = 6$
2. $x - 7 = -6$

Let's solve the first equation:

$x - 7 = 6$

Add 7 to both sides:

$x = 6 + 7$

$x = 13$

So, one solution is $x = 13$.

Now let's solve the second equation:

$x - 7 = -6$

Add 7 to both sides:

$x = -6 + 7$

$x = 1$

Thus, the second solution is $x = 1$.

Step 3: Verify the Solutions

It's always a good idea to check our solutions to make sure they're correct. Let's plug each value of x back into the original equation:

For $x = 13$:

$(13 - 7)^2 = 36$

$(6)^2 = 36$

$36 = 36$

This solution is correct!

For $x = 1$:

$(1 - 7)^2 = 36$

$(-6)^2 = 36$

$36 = 36$

This solution is also correct!

Conclusion

Alright, we've successfully solved the equation $(x-7)^2 = 36$! We found that the values of x are 13 and 1. This problem demonstrates a fundamental technique in solving quadratic equations, especially those in the form of a squared term equal to a constant. By taking the square root of both sides and considering both positive and negative roots, we can efficiently find the solutions.

Remember, understanding quadratic equations is crucial for many areas of math and science. Keep practicing, and you'll become a pro at solving these types of problems. Whether you're using factoring, completing the square, the quadratic formula, or taking the square root, each method offers a unique approach to tackling quadratic equations.

So, the correct values of x are 13 and 1. Great job, guys! Keep up the awesome work, and let's keep exploring the fascinating world of mathematics together.

Practice Problems

To solidify your understanding, try solving these similar equations:

1. $(x + 3)^2 = 25$
2. $(x - 5)^2 = 49$
3. $(2x - 1)^2 = 9

Solving these problems will help you become more comfortable with the method we discussed today. Remember to always check your solutions to ensure accuracy. Happy solving!

Further Exploration

If you're interested in diving deeper into quadratic equations, consider exploring the following topics:

• The discriminant ($b^2 - 4ac$) and its role in determining the nature of the roots.
• Graphing quadratic equations and understanding parabolas.
• Applications of quadratic equations in real-world scenarios.

By exploring these areas, you'll gain a more comprehensive understanding of quadratic equations and their significance in mathematics and beyond. Keep your curiosity alive, and always be eager to learn more!